Unformatted text preview: Q . (ii) Prove that − I is the only element in Q of order 2, and that all other elements M ±= I satisfy M 2 = − I . Solution. Straightforward multiplication. (iii) Show that Q has a unique subgroup of order 2, and it is the center of Q . Solution. A group of order 2 must be a cyclic group generated by an element of order 2. It is shown, in part (i), that − I is the only element of order 2. It is clear that h− I i ≤ Z ( Q ) , for scalar matrices commute with every matrix. On the other hand, for every element M ±= ² I , there is N ∈ Q with M N ±= N M . (iv) Prove that h− I i is the center Z ( Q ) . Solution. For each M ∈ Q but not in h− I i , there is a matrix M ∈ Q with M M ±= M M . 2.87 Prove that the quaternions Q and the dihedral group D 8 are nonisomorphic groups of order 8....
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- Fall '11
- Vector Space, operation matrix multiplication, Solution. Note ﬁrst